Kenneth Kuttler

280 CHAPTER 12. SERIES AND TRANSFORMS

It may be that the study of Fourier series and their convergence drove the developmentof real analysis during the nineteenth century as much as any other topic. Dirichlet did hiswork before Riemann gave the best description of the integral and it was Riemann whogave a general theory of integration including piecewise continuous functions. Cauchy’sintegral for continuous functions was the current state of the art at the time of Dirichlet.Thus the theorem about to be presented uses a more sophisticated theory of integrationthan that which was available then.

Recall limt→x+ f (t)≡ f (x+) , and limt→x− f (t)≡ f (x−).

Theorem 12.2.1 Let f be periodic of period 2π which is in R([−π,π]). Suppose atsome x, f (x+) and f (x−) both exist and that the Dini conditions hold which are that forsmall positive y,

| f (x− y)− f (x−)| ≤ Kyθ ,0 < θ ≤ 1, | f (x+ y)− f (x+)| ≤ Kyθ ,0 < θ ≤ 1

for 0 < y≤ δ , δ > 0. Then

limn→∞

Sn f (x) =f (x+)+ f (x−)

2. (12.5)

Proof: Sn f (x) =∫

−πDn (x− y) f (y)dy. Change variables x− y→ y and use the peri-

odicity of f and Dn along with the formula for Dn (y) to write this as

Sn f (x) =∫

−π

Dn (y) f (x− y)dy =∫

0Dn (y) f (x− y)dy+

∫ 0

−π

Dn (y) f (x− y)dy

=∫

0Dn (y) [ f (x− y)+ f (x+ y)]dy

=∫

1π

sin((

n+ 12

)y)

sin( y

) [f (x− y)+ f (x+ y)

]dy. (12.6)

since∫

−πDn (y)dy = 1,Sn f (x)− f (x+)+ f (x−)

2 =

∫π

1π

sin((

n+ 12

)y)

sin( y

) [f (x− y)+ f (x+ y)

2− f (x+)+ f (x−)

]dy

=∫

1π

sin((

n+12

)y)[

f (x− y)− f (x−)+ f (x+ y)− f (x+)

2sin( y

) ]dy

+∫

2π

sin((

n+12

)y)

y/2sin( y

) [ f (x− y)− f (x−)+ f (x+ y)− f (x+)

]dy (12.7)

In the first integral of 12.7, the function in [ ] is in L1 ([δ ,π]) because sin(y/2) isbounded away from 0. Therefore, the first of these integrals converges to 0 by the RiemannLebesgue theorem. In the second integral, |[ ]| ≤ Ky1−θ and y/2

sin( y2 )

is bounded on [0,δ ] so

this function multiplying sin((

n+ 12

)y)

is in L1 (0,δ ) . Therefore, the second integral alsoconverges to 0.

The following corollary is obtained immediately from the above proof with minor mod-ifications.

280 CHAPTER 12. SERIES AND TRANSFORMSIt may be that the study of Fourier series and their convergence drove the developmentof real analysis during the nineteenth century as much as any other topic. Dirichlet did hiswork before Riemann gave the best description of the integral and it was Riemann whogave a general theory of integration including piecewise continuous functions. Cauchy’sintegral for continuous functions was the current state of the art at the time of Dirichlet.Thus the theorem about to be presented uses a more sophisticated theory of integrationthan that which was available then.Recall lim,_,,.+ f (t) = f (x+), and lim,_,,— f (t) = f (x—).Theorem 12.2.1. Let f be periodic of period 2m which is in R ([—2, ]). Suppose atsome x, f (x+) and f (x—) both exist and that the Dini conditions hold which are that forsmall positive y,lf (x—y) —f (x-)| < Ky®,0< @ <1, |f(x+y) —f (x+)| < Ky®,0<@<1for0<y<6,65>0. Thenlim S,,f (x) = Ft) + fo) (12.5)n—yoo 2Proof: S,f (x) = [7,Dn(x—y) f (v) dy. Change variables x — y — y and use the peri-odicity of f and D, along with the formula for D, (y) to write this asDaly) fe-y)dy= ["Dalo) fee-yav | Daly) Fler) dyeTSf(s) =f—1= [ Proyife-») + F e+ y)lay= [Lanter )y) [FEF ay (12.6)0 FW1sin (3) 2since fn, Pn (y) dy = 1, Sy f (x) - fet )tie-) =[reer ee f(t) +f (x)0m sin(3) 2 — 2 dy_ *Lsn((n+4)») [feet ret ri dysin 5+ [°2sin((n+4)y) Ee . [Pe Foo APs) FO) (12.7)In the first integral of 12.7, the function in [ ] is in L'({6,2]) because sin(y/2) isbounded away from 0. Therefore, the first of these integrals converges to 0 by the RiemannLebesgue theorem. In the second integral, | ]| < Ky!~® and = a is bounded on [0, 5] sosin( >this function multiplying sin ((n + 4) y) is in L' (0,5). Therefore, the second integral alsoconverges to0. ffThe following corollary is obtained immediately from the above proof with minor mod-ifications.